By Aaron Marcus

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**Sample text**

31) the solution for w becomes (as each individual root must have an arbitrary constant A associated with it) : w = Aiexpß(l + i)x + A2expß(l — i)x + 4 3 exp — ß(l + i)x+A4exp —ß(l — i)x in which Ai to A4 are complex arbitrary constants. The above equation may be reduced to the following form on substituting cos ßx = (exp ißx + exp — ißx)j2 and sin ßx = (exp ißx—exp — ißx)\2i whence w = exp ßx[C1 cos ßx + C2 sin ßx] ^ -_ + exp — ßx\_C3 cos ßx + C4 sin ßx] in which Cl9 C2, C3, C4 are real arbitrary constants, and this is the required complementary function.

5). 8) By considering the strain and curvature displacement equations it will be shown that both Μφ and Nx may be expressed as functions of the radial displacement w. 8) may be expressed as a partial differential equation in w. 3. Strain Displacement Equations of the Middle Surface Any point on the middle surface of the shell element will suffer displacements w, v and w in the directions of x, y and z respectively. The membrane strain εί of the middle surface in the direction of 48 Linear Elastic Theory of Thin Shells x can be derived in exactly the same manner as in the case of the closed cylindrical shell.

3. Strain Displacement Equations of the Middle Surface Any point on the middle surface of the shell element will suffer displacements w, v and w in the directions of x, y and z respectively. The membrane strain εί of the middle surface in the direction of 48 Linear Elastic Theory of Thin Shells x can be derived in exactly the same manner as in the case of the closed cylindrical shell. 17) except that du/dx becomes du/dx since u is now a function of both x and φ. 9) ßi = τ - ox The membrane strain ε2 in the γ = αφ direction, however, demands closer examination.